Mathematics – Algebraic Geometry
Scientific paper
2006-10-28
SIGMA 3:066,2007
Mathematics
Algebraic Geometry
This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Sym
Scientific paper
10.3842/SIGMA.2007.066
We propose the graph description of Teichm\"uller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichm\"uller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to $A_n$ and $D_n$ algebras and discuss briefly the relation to the Thurston theory.
Chekhov Leonid
No associations
LandOfFree
Teichmüller Theory of Bordered Surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Teichmüller Theory of Bordered Surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Teichmüller Theory of Bordered Surfaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-448392